Regularity of $p(·)$-superharmonic functions, the Kellogg property and semiregular boundary points

Adamowicz, Tomasz; Björn, Anders; Björn, Jana

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1131-1153.

We study various boundary and inner regularity questions for $p(·)$-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for $p(·)$-harmonic functions into three disjoint classes: regular, semiregular and strongly irregular points. Regular and especially semiregular points are characterized in many ways. The discussion is illustrated by examples.Along the way, we present a removability result for bounded $p(·)$-harmonic functions and give some new characterizations of $W_0^{1,p(·)}$ spaces. We also show that $p(·)$-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.

Classification : 35J67, 31C45, 46E35
Mots clés : Comparison principle, Kellogg property, lsc-regularized, Nonlinear potential theory, Nonstandard growth equation, Obstacle problem, $ p(\cdot )$-harmonic, Quasicontinuous, Regular boundary point, Removable singularity, Semiregular point, Sobolev space, Strongly irregular point, $ p(\cdot )$-superharmonic, $ p(\cdot )$-supersolution, Trichotomy, Variable exponent

[1] E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), 213 -259 | MR | Zbl

[2] E. Acerbi, G. Mingione, Gradient estimates for the $p\left(x\right)$-Laplacean system, J. Reine Angew. Math. 584 (2005), 117 -148 | MR | Zbl

[3] T. Adamowicz, P. Hästö, Mappings of finite distortion and PDE with nonstandard growth, Int. Math. Res. Not. 2010 (2010), 1940 -1965 | MR | Zbl

[4] T. Adamowicz, P. Hästö, Harnack's inequality and the strong $p(·)$-Laplacian, J. Differ. Equ. 250 (2011), 1631 -1649 | MR | Zbl

[5] Yu.A. Alkhutov, O.V. Krasheninnikova, Continuity at boundary points of solutions of quasilinear elliptic equations with a nonstandard growth condition, Izv. Ross. Akad. Nauk Ser. Mat. 68 no. 6 (2004), 3 -60 , Izv. Math. 68 (2004), 1063 -1117 | MR | Zbl

[6] D.H. Armitage, S.J. Gardiner, Classical Potential Theory, Springer, London (2001) | MR

[7] A. Björn, A regularity classification of boundary points for p-harmonic functions and quasiminimizers, J. Math. Anal. Appl. 338 (2008), 39 -47 | MR | Zbl

[8] A. Björn, J. Björn, Boundary regularity for p-harmonic functions and solutions of the obstacle problem, J. Math. Soc. Jpn. 58 (2006), 1211 -1232 | MR | Zbl

[9] A. Björn, J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts Math. vol. 17 , European Math. Soc., Zürich (2011) | MR | Zbl

[10] A. Björn, J. Björn, U. Gianazza, M. Parviainen, Boundary regularity for degenerate and singular parabolic equations, in preparation.

[11] A. Björn, J. Björn, N. Shanmugalingam, The Dirichlet problem for p-harmonic functions on metric spaces, J. Reine Angew. Math. 556 (2003), 173 -203 | MR | Zbl

[12] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 no. 4 (2006), 1383 -1406 | MR | Zbl

[13] L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lect. Notes Math. vol. 2017 , Springer, Berlin, Heidelberg (2011) | MR | Zbl

[14] X.-L. Fan, Global $C^{1,\alpha }$ regularity for variable exponent elliptic equations in divergence form, J. Differ. Equ. 235 (2007), 397 -417 | Zbl

[15] X.-L. Fan, Y.Z. Zhao, Q.-H. Zhang, A strong maximum principle for $p\left(x\right)$-Laplace equations, Chin. Ann. Math., Ser. A 24 (2003), 495 -500 , Chin. J. Contemp. Math. 24 (2003), 277 -282

[16] P. Harjulehto, P. Hästö, M. Koskenoja, T. Lukkari, N. Marola, An obstacle problem and superharmonic functions with nonstandard growth, Nonlinear Anal. 67 (2007), 3424 -3440 | MR | Zbl

[17] P. Harjulehto, P. Hästö, V. Latvala, O. Toivanen, The strong minimum principle for quasisuperminimizers of non-standard growth, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28 (2011), 731 -742 | Numdam | MR | Zbl

[18] P. Harjulehto, P. Hästö, Ú.V. Lê, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), 4551 -4574 | MR | Zbl

[19] P. Harjulehto, J. Kinnunen, T. Lukkari, Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl. 2007 (2007) | EuDML | MR | Zbl

[20] P. Harjulehto, V. Latvala, Fine topology of variable exponent energy superminimizers, Ann. Acad. Sci. Fenn. Math. 33 (2008), 491 -510 | EuDML | MR | Zbl

[21] J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York (2001) | MR | Zbl

[22] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover, Mineola, NY (2006) | MR | Zbl

[23] E. Henriques, Regularity for the porous medium equation with variable exponent: the singular case, J. Differ. Equ. 244 (2008), 2578 -2601 | MR | Zbl

[24] T. Kilpeläinen, A remark on the uniqueness of quasi continuous functions, Ann. Acad. Sci. Fenn. Math. 23 no. 1 (1998), 261 -262 | EuDML | MR | Zbl

[25] V. Latvala, T. Lukkari, O. Toivanen, The fundamental convergence theorem for $p(·)$-superharmonic functions, Potential Anal. 35 (2011), 329 -351 | MR | Zbl

[26] H. Lebesgue, Sur des cas d'impossibilité du problème de Dirichlet ordinaire, in Vie de la société, Bull. Soc. Math. Fr. 41 (1913), 1 -62

[27] J. Lukeš, J. Malý, On the boundary behaviour of the Perron generalized solution, Math. Ann. 257 (1981), 355 -366 | EuDML | MR | Zbl

[28] T. Lukkari, Singular solutions of elliptic equations with nonstandard growth, Math. Nachr. 282 (2009), 1770 -1787 | MR | Zbl

[29] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lect. Notes Math. vol. 1748 , Springer, Berlin (2000) | MR | Zbl

[30] S. Zaremba, Sur le principe de Dirichlet, Acta Math. 34 (1911), 293 -316 | JFM | MR